His chief security minister, Wiranto, also pledged a "complete and fair" investigation into incidents in East Java that triggered the protests and in Papua, saying the situation in Papua had been contained.
A separatist movement has simmered for decades in Papua, where there have been frequent complaints of rights abuses by Indonesian security forces.
The latest protest appears to have been sparked off by the detention of Papuan students in the East Java city of Surabaya accused of bending a flagpole in front of a dormitory during Independence Day celebrations on Saturday, activists say.
Police fired tear gas into the dormitory before arresting 43 students, Albert Mungguar, an activist, told a news conference on Sunday. He said the students, who were released the same day without charge, had been called "monkeys" during the operation.
Thousands of Papuans protested in the cities of Manokwari and Sorong, blocking streets by burning tyres and tree branches.
Papuans were angry because of "the extremely racist words used by East Java people, the police and military", Papua Governor Lukas Enembe told broadcaster TVone.
In Manokwari, the capital of West Papua province, protesters set fire to parliament and office buildings, pulled down power poles and burnt vehicles, Deputy Governor Mohamad Lakotani said by telephone.
Protesters entered the airport in Sorong and destroyed some facilities, state news agency Antara said.
Widodo called for calm in Papua and urged people not to damage public facilities.
"It's okay to be emotional, but it's better to be forgiving," he told reporters at the presidential palace, a television broadcast showed. "Patience is also better."
Lakotani, who met protesters in Manokwari, said Papuans demanded an apology for the slur against the students, as well as protection for anyone studying across the archipelago.
"We apologise because this does not represent the voice of the people of East Java," the province's governor, Khofifah Indar Parawansa, said in a televised statement and called the slur "someone's personal outburst of emotion".
Large crowds also took to the streets of Jayapura, the capital of Papua province, though the protest appeared peaceful in television images.
Papua police spokesman Ahmad Kamal said by telephone 500 people were involved in the Jayapura demonstration.
"It's been a while since I saw West Papuans this angry," Veronica Koman, a human rights lawyer who focuses on Papua, said on Twitter. "The liberation movement is entering a new chapter."
Koman posted videos on Twitter of protesters crying, "Free Papua" and a group of teenagers carrying a Morning Star flag, a banned symbol used by independence supporters.
Papua and West Papua provinces, the resource-rich western part of the island of New Guinea, make up a former Dutch colony that was incorporated into Indonesia after a widely criticised UN-backed referendum in 1969.
President Joko Widodo has sought to ease tension in the restive region with steps such as building the Trans Papua highway to spur economic activity and boost welfare.
However, unrest has persisted and separatists killed a group of construction workers in December 2018, triggering a military crackdown that displaced thousands in the Nduga area.